No 10 said it hoped the new treaty with Germany could be agreed by early next year.
Although detailed plans have not been disclosed, discussions about improving market access are expected to focus on areas such as helping firms certificate their products, and providing more information about tenders.
Downing Street added it would build on a defence pact between the two countries already being negotiated, which is due to be finalised in the autumn.
That agreement, announced last month, saw the two countries pledge to buy more military equipment together and make it easier for each other’s armies to use it, as well increase co-operation in areas such as cyber warfare.
Sir Keir has vowed to forge a closer economic relationship with Europe, including a “much better” deal on trade than the one negotiated by Boris Johnson in late 2020.
His Labour government wants to strike deals with the EU to reduce border checks on food products, lessen paperwork for touring artists, and boost recognition of work qualifications, making it easier for some professionals to work abroad.
It also wants to broker a security pact with the EU, as well as a new returns agreement for failed asylum seekers.
It remains unclear whether Brussels would entertain major changes to the UK’s existing Brexit trade deal, which is due to be reviewed in 2026.